142 ANALYTICAL MECHANICS 115. Center of Mass of Continuous Bodies.—When the particles form a continuous body we can replace the summation signs of equation (I') by integration signs and obtain the following expressions for the coordinates of the center of mass: x=^ r' dm . y dm y—r—> I dm Jo r, Jo z dm r Jo dm (I) where m is the mass of the body. ILLUSTRATIVE EXAMPLES. 1. Find the center of mass of the parabolic lamina bounded by the curves ?/ = 2 px and x = a, Fig. 74. Obviously the center of mass lies on the re-axis. Therefore we need to 0 4a '----^ FIG. 74. FIG. 75. * In general if y is a function of x then the average value of y between the limits Xi and x* is given by the relation: y = —--— f^ydx.